For identity (3), recall that in any Segal space \(X\)  there is a notion of when a morphism \(f \in X_1\)  is invertible (or a `homotopy equivalence' in the language of [1]}).
One defines the space \(X_{weq}\)  of homotopy equivalences in \(X\)  to be the union of those connected components of \(X_1\)  that contain invertible morphisms (by [1]}, if \(X \in {{SS}\hspace{-0.56917pt}\mathrm {p}}\)  is fibrant, then a connected component of \(X_1\)  contains a homotopy equivalence if and only if it consists purely of homotopy equivalences).
For any Segal space, the degeneracy morphism \(s_0 \colon X_0 \rightarrow X_1\)  factors as
\(\begin{tikzcd}& X_{weq} [d, hookrightarrow, "\iota _X"]\\X_0 [ur, "\widehat{s_0}"] [r, "s_0"^{\prime }] & X_1\end{tikzcd}\) 
